Hyperbolic lengths of geodesics surrounding two punctures

Abstract

For the plane regions Ω 1 = { | z | > R , z ≠ 0 , 1 } {\Omega _1} = \left \{ {\left | z \right | > R,z \ne 0,1} \right \} with R > 1 R > 1 , and Ω 2 = C ∖ { 0 , 1 , p } {\Omega _2} = {\mathbf {C}} \setminus \left \{ {0,1,p} \right \} with | p | = R > 1 \left | p \right | = R > 1 , we describe, as R → ∞ R \to \infty , the hyperbolic lengths of the geodesies surrounding 0 and 1. Upper and lower bounds for the lengths are also stated, and these results are used to obtain inequalities, which are precise in a certain sense, for the length of the geodesic surrounding 0 and 1 in an arbitrary plane region Ω \Omega satisfying Ω 1 ⊂ Ω ⊂ Ω 2 {\Omega _1} \subset \Omega \subset {\Omega _2} .